altera_dspbook.pdf

altera_dspbook.pdf

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1DigitalSignalProcessing:

APracticalGuideByMichaelParker,AlteraCorporationIntroductionThisbookisintendedforthosewhoworkinorprovidecomponentsforindustriesthatusedigitalsignalprocessing（DSP）.Thereisawidevarietyofindustriesthatutilizethistechnology.WhiletheengineerswhoimplementapplicationsusingDSPmustbeveryfamiliarwiththetechnology,therearemanyotherswhocanbenefitfromabasicknowledgeofitsfundamentalprincipals,whichisthegoalofthisbook-toprovideabasictutorialonDSP.Mathematicswillbeminimizedandintuitiveunderstandingmaximized.Thisbookwillde-mistifymanydifficultconceptslikesampling,aliasing,imaginarynumbers,frequencyresponse,etc.,usingeasytounderstandexamples,whileprovidinganoverviewofthefunctionsandimplementationusedinseveralDSPintensiveapplications.ThisisnotablackboardofferingofequationsasanexplanationonDSP.Thereaderneedonlybecomfortablewithhighschoollevelmathskills.Keyconceptswillbeemphasized.AfteracompletereadingyouwilllikelybeabletotalkintelligentlywithothersinvolvedinDSPcentricindustriesandunderstandmanyofitsfundamentalconcepts.OK,letsgetstartedWhatisDSP?

DSPisperformingoperationsonadigitalsignalusingadigitalsemiconductordevice.Mostcommonly,multipliersandaddersareused.Ifyoucanmultiplyandadd,youcanunderstandthetechnology.Digitalcircuitshavebecomeprogressivelycheaperandfaster,aswellasduetotheinherentadvantagesofrepeatability,toleranceandconsistencythatdigitalcircuitshavecomparedtoanalogcircuits.Ifthesignalisnotinadigitalform,thenitmustfirstbyconvertedordigitized.Adevicecalledananalogtodigitalconverter（ADC）isused.Iftheoutputsignalneedstobeanalog,thenisconvertedbackusingadigitaltoanalogconverter（DAC）.Aspromised,mathematicswillbeminimized,butcannotbeeliminatedaltogether.Somebasictrigonometryanduseofcomplexnumbersisunavoidable（anearlychapterintroducestheseusingsimpleexamples）.Thereisalsooneappendixsectionwhereverybasiccalculusisused,butthisisnotessentialtotheoverallunderstanding.Asthis7partseriesisonlyintendedasanoverviewofthecompletebookanditschapters,youwilllikelyhavetoactuallypurchasethisbook（goodformeandthepublisher）togainacompleteunderstanding.ItismygoalinthisseriesofarticlestogiveaclearandconcisesummaryofthebookfollowingitsTableofContents.ThisfirstpartwillcoverChapters12Chapter1:

NumericalRepresentationToprocessasignaldigitally,itmustberepresentedinadigitalformat.Thereareanumberofdifferentwaystorepresentnumbers,andthisgreatlyaffectsboththeresultandtheamountofcircuitresourcesrequiredtoperformagivenoperation.ThischapterfocusesonimplementingDSPandyoudonotneedtounderstandDSPfundamentalconcepts.Digitalelectronicsoperateonbitsthatformbinarywords.Thebitscanberepresentedasbinary,decimal,octalorhexadecimalorotherform.Thesebinarynumberscanbeusedtorepresent“real”numbers.TherearetwobasictypesofarithmeticusedinDSP,floatingpointorfixedpoint.Fixedpointnumbershaveafixeddecimalpointaspartofthenumber（i.e.1234,12.34or0.1234）.Floatingpointisanumberwithanexponent（i.e.1,200,000wouldbeexpressedas1.2x106）.MostofthediscussionfocusesonfixedpointnumbersthataremostcommonlyfoundinDSPapplicationsandashortdiscussiononfloatingpointnumbers.DSPusessignednumbersthatarebothpositiveandnegativenumbers,buthowarenegativenumbersrepresented?

Insignedfixedpointarithmetic,thebinarynumberrepresentationsincludeasign,aradixordecimalpoint,andthemagnitude.Thesignindicateswhetherthenumberispositiveornegative,andtheradix（alsocalleddecimal）pointseparatestheintegerandfractionalpartsofthenumber.Thesignisnormallydeterminedbytheleftmost,ormostsignificantbit（MSB）.Theconventionisazerousedforpositive,andonefornegative.Thereareseveralformatstorepresentnegativenumbers,butthealmostuniversalmethodisknownas“2scomplement”.Thismethodisdiscussedindetailinthebook.Fixedpointnumbersareusuallyrepresentedaseitherintegerorfractional.Anintegerrepresentation,thedecimalpointistotherightoftheleastsignificantbit（LSB）,andthereisnofractionalpartinthenumber.Foran8bitnumber,therangewhichcanberepresentedisfrom-128to+127withincrementsof1.Infractionalrepresentation,thedecimalpointisoftenjusttotherightoftheMSB,whichisalsothesignbit.Foran8bitnumber,therangewhichcanberepresentedisfrom-1to+127/128（almost+1）withincrementsof1/128.Inthebook,thischapterpresentsnumeroustables,witheachgivingequivalentbinaryandhexadecimalnumbers.ThebookwillcoverIntegerFixedPointRepresentation,FractionalFixedPointRepresentation,andFloatingPointRepresentationindepth.Chapter2:

ComplexNumbersandExponentialsComplexnumbershaveprobablybeenforgottenbymostofus.TheyareimportantindigitalcommunicationsandDSP,soyoullneedtoresurrect3Acomplexnumberhasa“real”and“imaginary”part,andtheimaginarypartisthesquarerootofanegativenumber,whichisreallyanon-existentnumber.Soundweird,wellitstechnicallytrue,soitwillbeexaminedinamuchmoreintuitivewayinthebook.ComplexnumbersareneededtodefineatwodimensionalnumberplanetounderstandDSP.Thetraditionalnumberlineextendsfromplusinfinitytominusinfinity,alongasingleline.TorepresentmanyoftheconceptsinDSP,twodimensionsareneeded.Thisrequirestwoorthogonalaxes（i.e.thehorizontallineistherealnumberline,theverticallineistheimaginaryline）.Allimaginarynumbersareprefacedby“j”,whicharedefinedasthe-1.ThisistheessenceofthiswholechapterasshowninFigure1.Figure1.AComplexNumberPlaneAnycomplexnumberZhasarealandimaginarypart,andisexpressedasX+jY,orjustX+jY.ThevalueofXandYforanypointisdeterminedbythedistanceonemusttravelinthedirectionofeachaxistoarriveatthepoint.Itcanalsobevisualizedasapointonthecomplexnumberplane,orasavectororiginatingattheoriginandterminatingatthepoint.Therehastobeawaytokeeptracktheverticalandhorizontalcomponents.Thatswherethe“j”comesin.ComplexAddition,SubtractionandMultiplicationexplanationsandexampleswillbegiveninthebook,aswellasin-depthoverviewsofPolarRepresentationandComplexMultiplicationusingPolarRepresentationexamples.Twopointswillbedefined,Z1andZ2.Z1=R1angle

（1）Z2=R2angle

（2）Z1Z2=（R1R2）angle（1+2）4Whatthismeansisthatwithanytwocomplexnumbers,themagnitude,ordistancefromtheorigintotheradius,getsmultipliedtogethertoformthenewmagnitude.Exampleswillbegiveninthebook.ComplexConjugateThisisthelastexamplethatwillbecoveredinthebook.Thisisaspecialcaseandwillbeexplainedwhy.Imaginarynumbersareusedtoformcomplexnumbers.Theyarereallynotsocomplex,andimaginaryisreallyaverymisleadingdescription.Whatwillbeexplainedistohowtocreateatwodimensionalnumberplaneanddefineasetofexpandedarithmeticrulestomanipulatethenumbersinit.Nextthecomplexexponentialwhichissimplytheunitcircle（radius=1）onthecomplexnumberplanewillbeexplained.Thelastpartofthischapterinvolvesmeasuringanglesinradians,whichisseeneverywhereinDSP.Theanglemeasurementinradiansisbasedupon,whichisanumberdefinedtohaveavalueofabout3.141592（itactuallyisanirrationalnumber,withinfinitenumberofdigits,like1/3=0.3333.）.Ittakesexactly2radianstodescribeafullcircle.Thissameconceptwillbeexaminedlaterinsamplingtheory,whereeverythingtendstowraparoundorbehaveperiodically.Wecanvisualizethisastravelingeitherclockwise（negativerotation）orcounterclockwise（positiverotation）aroundthecircle.ThereisonemoreDSPconventiontobeawareof.Therealcomponent（Xwasusedearlier）itisusuallycalledthe“I”or“in-phase”component,andtheimaginarycomponent（usedY）isusuallyreferredtoasthe“Q”or“quadrature”component.InmanyDSPalgorithms,thedigitalsignalprocessingmustbeperformedsimultaneouslyonbothIandQdatastreams,whichsimplyrepresentsthesignalsmovement,overtime,withinthetwodimensionsofthecomplexnumberplane.Chapter3:

Sampling,AliasingandQuantizationThestartingpointtounderstandDSPissampling,anditsaffectonthesignalofinterest.TheADCmeasuresthesignalatrapidintervals,calledsamples.Adigitalsignalproportionaltotheamplitudeoftheanalogsignalatthatinstantisoutput.Ifasignalissampledveryfastcomparedtohowrapidlythesignalischanging,afairlyaccuratesamplerepresentationofthesignalisobtained,butifthesampleistooslow,adistortedversionofthesignalisseen.Figures2and3aresamplingsoftwodifferentsinusoidalsignals.Theslowermovingsignal（lowerfrequency）canberepresentedaccuratelywiththeindicatedsamplerate,butthefastermovingsignal（higherfrequency）isnotaccuratelyrepresentedbythesamplerate.Infact,itactuallyappearstobeaslowmoving（lowfrequency）signal,asindicatedbythedashedline.Thisshowstheimportanceofsampling“fast”enoughforagivenfrequency5Figure2.Samplingalowfrequencysignal（arrowsindicatesampleinstants）Figure3.Samplingahighfrequencysignal（samesampleinstants）Thedashedbluelineshowshowthesampledsignalwillappearifthesampledotsandsmoothoutthesignalareconnected.Noticethatsincetheactual（redsolidline）signalischangingsorapidlybetweensamplinginstants,thismovementisnotapparentinthesampledversionofthesignal.Thesampledversionactuallyappearstobealowerfrequencysignalthantheactual